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Convergence of the parabolic Ginzburg-Landau equation to motion by mean curvature. (English) Zbl 1103.35038

The paper considers the initial-value problem for the multidimensional Ginzburg-Landau equation with real coefficients (but for a complex function). A theorem is rigorously proven that states that, asymptotically at \(t\to\infty\), the vorticity of the generic solution evolves in time according to the motion by mean curvature (in the special weak formulation introduced by K. A. Brakke [The motion of a surface by its mean curvature. Princeton, New Jersey: New Jersey University Press (1978; Zbl 0386.53047)], which allows one to encompass possible singularities of the solution). The only essential assumption used for the proof is a natural bound on the value of “energy” (in fact, Lyapunov’s functional) of the initial data.

MSC:

35K55 Nonlinear parabolic equations
35Q35 PDEs in connection with fluid mechanics
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
35B25 Singular perturbations in context of PDEs
35K15 Initial value problems for second-order parabolic equations
49Q20 Variational problems in a geometric measure-theoretic setting

Citations:

Zbl 0386.53047
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